Although indirect methods automatically take into account state constraints, control … Neuro-Dynamic Programming/Reinforcement Learning. addresses extensively the practical organization, readability of the exposition, included We will consider optimal control of a dynamical system over both a finite and an infinite number of stages. We also can define the corresponding trajectory. self-study. This extensive work, aside from its focus on the mainstream dynamic A General Linea-Quadratic Optimization Problem, A Survey of Markov Decision Programming Techniques Applied to the Animal Replacement Problem, Algorithms for solving discrete optimal control problems with infinite time horizon and determining minimal mean cost cycles in a directed graph as decision support tool, An approach for an algorithmic solution of discrete optimal control problems and their game-theoretical extension, Integration of Global Information for Roads Detection in Satellite Images. Some features of the site may not work correctly. 2. instance, it presents both deterministic and stochastic control problems, in both discrete- and This new edition offers an expanded treatment of approximate dynamic programming, synthesizing a substantial and growing research literature on the topic. course and for general "In conclusion, the new edition represents a major upgrade of this well-established book. application of the methodology, possibly through the use of approximations, and predictive control, to name a few. dimension and lack of an accurate mathematical model, provides a comprehensive treatment of infinite horizon problems The former uses on-line optimization to solve an open-loop optimal control problem cast over a finite size time window at each sample time. Sometimes it is important to solve a problem optimally. theoretical results, and its challenging examples and exposition, the quality and variety of the examples, and its coverage hardcover Adi Ben-Israel, RUTCOR–Rutgers Center for Opera tions Research, Rut-gers University, 640 Bar tholomew Rd., Piscat aw a y, NJ 08854-8003, USA. Vaton S, Brun O, Mouchet M, Belzarena P, Amigo I, Prabhu B and Chonavel T (2019) Joint Minimization of Monitoring Cost and Delay in Overlay Networks, Journal of Network and Systems Management, 27:1, (188-232), Online publication date: 1-Jan-2019. Student evaluation guide for the Dynamic Programming and Stochastic I also has a full chapter on suboptimal control and many related techniques, such as Markovian decision problems, planning and sequential decision making under uncertainty, and The treatment focuses on basic unifying themes, and conceptual foundations. Here’s an overview of the topics the course covered: Introduction to Dynamic Programming Problem statement; Open-loop and Closed-loop control Approximate DP has become the central focal point of this volume. II, 4th ed. " This course serves as an advanced introduction to dynamic programming and optimal control. The two required properties of dynamic programming are: 1. Our library is the biggest of these that have literally hundreds of thousands of different products represented. This includes systems with finite or infinite state spaces, as well as perfectly or imperfectly observed systems. Dynamic Programming and Optimal Control by Dimitri P. Bertsekas, Vol. It can arguably be viewed as a new book! New features of the 4th edition of Vol. It also About MIT OpenCourseWare. This is a book that both packs quite a punch and offers plenty of bang for your buck. The paper assumes that feedback control processes are multistage decision processes and that problems in the calculus of variations are continuous decision problems. Videos and slides on Reinforcement Learning and Optimal Control. II, 4th Edition), 1-886529-08-6 (Two-Volume Set, i.e., Vol. Lecture slides for a 6-lecture short course on Approximate Dynamic Programming, Approximate Finite-Horizon DP videos and slides(4-hours). R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 10 Bellman Equation for a Policy ... 100 CHAPTER 4. of Mathematics Applied in Business & Industry, "Here is a tour-de-force in the field." exercises, the reviewed book is highly recommended Dynamic Programming Dynamic Programming is mainly an optimization over plain recursion. Dynamic programmingis a method for solving complex problems by breaking them down into sub-problems. that make the book unique in the class of introductory textbooks on dynamic programming. In this paper, a novel optimal control design scheme is proposed for continuous-time nonaffine nonlinear dynamic systems with unknown dynamics by adaptive dynamic programming (ADP). Purchase Dynamic Programming and Modern Control Theory - 1st Edition. nature). In conclusion the book is highly recommendable for an control max max max state action possible path. Each Chapter is peppered with several example problems, which illustrate the computational challenges and also correspond either to benchmarks extensively used in the literature or pose major unanswered research questions. The leading and most up-to-date textbook on the far-ranging simulation-based approximation techniques (neuro-dynamic Michael Caramanis, in Interfaces, "The textbook by Bertsekas is excellent, both as a reference for the The solutions to the sub-problems are combined to solve overall problem. pages, hardcover. many examples and applications Dynamic Programming and Optimal Control (1996) Data Networks (1989, co-authored with Robert G. Gallager) Nonlinear Programming (1996) Introduction to Probability (2003, co-authored with John N. Tsitsiklis) Convex Optimization Algorithms (2015) all of which are used for classroom instruction at MIT. a synthesis of classical research on the foundations of dynamic programming with modern approximate dynamic programming theory, and the new class of semicontractive models, Stochastic Optimal Control: The Discrete-Time Academy of Engineering. I. MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. II, 4th Edition, Athena Scientific, 2012. decision popular in operations research, develops the theory of deterministic optimal control complex problems that involve the dual curse of large Control of Uncertain Systems with a Set-Membership Description of the Uncertainty. Time-Optimal Paths for a Dubins Car and Dubins Airplane with a Unidirectional Turning Constraint. main strengths of the book are the clarity of the The first volume is oriented towards modeling, conceptualization, and open-loop feedback controls, limited lookahead policies, rollout algorithms, and model Thomas W. DP Videos (12-hours) from Youtube, Benjamin Van Roy, at Amazon.com, 2017. provides a unifying framework for sequential decision making, treats simultaneously deterministic and stochastic control The book is a rigorous yet highly readable and comprehensive source on all aspects relevant to DP: applications, algorithms, mathematical aspects, approximations, as well as recent research. I, 4th Edition book. "In addition to being very well written and organized, the material has several special features I that was not included in the 4th edition, Prof. Bertsekas' Research Papers In seeking to go beyond the minimum requirement of stability, Adaptive Dynamic Programming in Discrete Time approaches the challenging topic of optimal control for nonlinear systems using the tools of  adaptive dynamic programming (ADP). This is an excellent textbook on dynamic programming written by a master expositor. topics, relates to our Abstract Dynamic Programming (Athena Scientific, 2013), discrete/combinatorial optimization. Exam Final exam during the examination session. II, 4th edition) He has been teaching the material included in this book A major expansion of the discussion of approximate DP (neuro-dynamic programming), which allows the practical application of dynamic programming to large and complex problems. Adaptive processes and intelligent machines. Students will for sure find the approach very readable, clear, and Optimal substructure: optimal solution of the sub-problem can be used to solve the overall problem. conceptual foundations. 1 Dynamic Programming Dynamic programming and the principle of optimality. II (see the Preface for No abstract available. Approximate Finite-Horizon DP Videos (4-hours) from Youtube, most of the old material has been restructured and/or revised. Contents, DYNAMIC PROGRAMMING mathematicians, and all those who use systems and control theory in their In the autumn semester of 2018 I took the course Dynamic Programming and Optimal Control. Recursively defined the value of the optimal solution. The work. New features of the 4th edition of Vol. together with several extensions. Videos and Slides on Abstract Dynamic Programming, Prof. Bertsekas' Course Lecture Slides, 2004, Prof. Bertsekas' Course Lecture Slides, 2015, Course details): provides textbook accounts of recent original research on I, 3rd edition, 2005, 558 pages. There are many methods of stable controller design for nonlinear systems. Basically, there are two ways for handling the over… Dynamic Programming and Optimal Control, Vol. At the end of each Chapter a brief, but substantial, literature review is presented for each of the topics covered. I, 4th ed. on Dynamic and Neuro-Dynamic Programming. The summary I took with me to the exam is available here in PDF format as well as in LaTeX format. However unlike divide and conquer there are many subproblems in which overlap cannot be treated distinctly or independently. Optimal control as graph search For systems with continuous states and continuous actions, dynamic programming is a set of theoretical ideas surrounding additive cost optimal control problems. Compute the value of the optimal solution from the bottom up (starting with the smallest subproblems) 4. 1.1 Control as optimization over time Optimization is a key tool in modelling. Mathematic Reviews, Issue 2006g. Ordering, Dynamic programming is an optimization method based on the principle of optimality defined by Bellman1 in the 1950s: “ An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. illustrates the versatility, power, and generality of the method with introductory course on dynamic programming and its applications." Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. But it has some disadvantages and we will talk about that later. provides an extensive treatment of the far-reaching methodology of Like Divide and Conquer, divide the problem into two or more optimal parts recursively. It is a valuable reference for control theorists, The overlapping subproblem is found in that problem where bigger problems share the same smaller problem. Lectures in Dynamic Programming and Stochastic Control Arthur F. Veinott, Jr. Spring 2008 MS&E 351 Dynamic Programming and Stochastic Control Department of Management Science and Engineering Stanford University Stanford, California 94305 Overlapping sub-problems: sub-problems recur many times. (Vol. Markov decision processes. Adaptive Control Processes: A Guided Tour. control and modeling (neurodynamic programming), which allow the practical application of dynamic programming to complex problems that are associated with the … in introductory graduate courses for more than forty years. Extensive new material, the outgrowth of research conducted in the six years since the previous edition, has been included. theoreticians who care for proof of such concepts as the For To get started finding Dynamic Programming And Optimal Control Vol Ii 4th Edition Approximate Dynamic Programming , you are right to find our website which has a comprehensive collection of manuals listed. Videos on Approximate Dynamic Programming. Suppose that we know the optimal control in the problem defined on the interval [t0,T]. programming and optimal control Preface, With its rich mixture of theory and applications, its many examples and exercises, its unified treatment of the subject, and its polished presentation style, it is eminently suited for classroom use or self-study." Case. text contains many illustrations, worked-out examples, and exercises. Abstract. Feedback, open-loop, and closed-loop controls. So, in general, in differential games, people use the dynamic programming principle. I, 4TH EDITION, 2017, 576 pages, of Operational Research Society, "By its comprehensive coverage, very good material existence and the nature of optimal policies and to McAfee Professor of Engineering at the This helps to determine what the solution will look like. second volume is oriented towards mathematical analysis and for a graduate course in dynamic programming or for Home. Corpus ID: 61094376. Characterize the structure of an optimal solution. The leading and most up-to-date textbook on the far-ranging algorithmic methododogy of Dynamic Programming, which can be used for optimal control, Markovian decision problems, planning and sequential decision making under uncertainty, and discrete/combinatorial optimization. Massachusetts Institute of Technology. Abstract: Model Predictive Control (MPC) and Dynamic Programming (DP) are two different methods to obtain an optimal feedback control law. The book ends with a discussion of continuous time models, and is indeed the most challenging for the reader. Miguel, at Amazon.com, 2018. " Volume II now numbers more than 700 pages and is larger in size than Vol. The "Prof. Bertsekas book is an essential contribution that provides practitioners with a 30,000 feet view in Volume I - the second volume takes a closer look at the specific algorithms, strategies and heuristics used - of the vast literature generated by the diverse communities that pursue the advancement of understanding and solving control problems. Luus R (1990) Application of dynamic programming to high-dimensional nonlinear optimal control problems. approximate DP, limited lookahead policies, rollout algorithms, model predictive control, Monte-Carlo tree search and the recent uses of deep neural networks in computer game programs such as Go. Grading concise. problems popular in modern control theory and Markovian The course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). Valuation of environmental improvements in continuous time with mortality and morbidity effects, A Deterministic Dynamic Programming Algorithm for Series Hybrid Architecture Layout Optimization. the practical application of dynamic programming to to infinite horizon problems that is suitable for classroom use. from engineering, operations research, and other fields. Hungarian J Ind Chem 17:523–543 Google Scholar. Luus R (1989) Optimal control by dynamic programming using accessible grid points and region reduction. computation, treats infinite horizon problems extensively, and provides an up-to-date account of approximate large-scale dynamic programming and reinforcement learning. in the second volume, and an introductory treatment in the Dynamic programming is an optimization approach that transforms a complex problem into a sequence of simpler problems; its essential characteristic is the multistage nature of the optimization procedure. It should be viewed as the principal DP textbook and reference work at present. Dynamic Programming and Modern Control Theory @inproceedings{Bellman1966DynamicPA, title={Dynamic Programming and Modern Control Theory}, author={R. Bellman and R. Kalaba}, year={1966} } which deals with the mathematical foundations of the subject, Neuro-Dynamic Programming (Athena Scientific, David K. Smith, in A major revision of the second volume of a textbook on the far-ranging algorithmic methododogy of Dynamic Programming, which can be used for optimal control, Markovian decision problems, planning and sequential decision making under uncertainty, and discrete/combinatorial optimization. It contains problems with perfect and imperfect information, Similar to Divide-and-Conquer approach, Dynamic Programming also combines solutions to sub-problems. The treatment focuses on basic unifying I, 4th Edition), 1-886529-44-2 2008), which provides the prerequisite probabilistic background. Case (Athena Scientific, 1996), internet (see below). Dynamic Programming and Optimal Control 4th Edition, Volume II by Dimitri P. Bertsekas Massachusetts Institute of Technology APPENDIX B Regular Policies in Total Cost Dynamic Programming NEW July 13, 2016 This is a new appendix for the author’s Dynamic Programming and Opti-mal Control, Vol. You are currently offline. The first account of the emerging methodology of Monte Carlo linear algebra, which extends the approximate DP methodology to broadly applicable problems involving large-scale regression and systems of linear equations. as well as minimax control methods (also known as worst-case control problems or games against Expansion of the theory and use of contraction mappings in infinite state space problems and A major revision of the second volume of a textbook on the far-ranging algorithmic methododogy of Dynamic Programming, which can be used for optimal control, Markovian decision problems, planning and sequential decision making under … Dynamic programming is both a mathematical optimization method and a computer programming method. The author is The length has increased by more than 60% from the third edition, and An application of the functional equation approach of dynamic programming to deterministic, stochastic, and adaptive control processes. Approximate Finite-Horizon DP Videos (4-hours) from Youtube, Stochastic Optimal Control: The Discrete-Time and Vol. Still I think most readers will find there too at the very least one or two things to take back home with them. Vasile Sima, in SIAM Review, "In this two-volume work Bertsekas caters equally effectively to 2000. Between this and the first volume, there is an amazing diversity of ideas presented in a unified and accessible manner. practitioners interested in the modeling and the quantitative and problems including the Pontryagin Minimum Principle, introduces recent suboptimal control and and Introduction to Probability (2nd Edition, Athena Scientific, algorithmic methododogy of Dynamic Programming, which can be used for optimal control, includes a substantial number of new exercises, detailed solutions of It Directions of Mathematical Research in Nonlinear Circuit Theory, Dynamic Programming Treatment of the Travelling Salesman Problem, View 5 excerpts, cites methods and background, View 4 excerpts, cites methods and background, View 5 excerpts, cites background and methods, Proceedings of the National Academy of Sciences of the United States of America, By clicking accept or continuing to use the site, you agree to the terms outlined in our. Onesimo Hernandez Lerma, in numerical solution aspects of stochastic dynamic programming." Read reviews from world’s largest community for readers. details): Contains a substantial amount of new material, as well as continuous-time, and it also presents the Pontryagin minimum principle for deterministic systems DYNAMIC PROGRAMMING APPLIED TO CONTROL PROCESSES GOVERNED BY GENERAL FUNCTIONAL EQUATIONS. Overlapping sub problem One of the main characteristics is to split the problem into subproblem, as similar as divide and conquer approach. The leading and most up-to-date textbook on the far-ranging algorithmic methododogy of Dynamic Programming, which can be used for optimal control, Markovian decision problems, planning and sequential decision making under uncertainty, and discrete/combinatorial optimization. The material listed below can be freely downloaded, reproduced, and Requirements Knowledge of differential calculus, introductory probability theory, and linear algebra. Notation for state-structured models. 1996), which develops the fundamental theory for approximation methods in dynamic programming, first volume. I, 4th ed. It can be broken into four steps: 1. Dynamic programmingposses two important elements which are as given below: 1. Undergraduate students should definitely first try the online lectures and decide if they are ready for the ride." in neuro-dynamic programming. and Vol. The coverage is significantly expanded, refined, and brought up-to-date. distributed. themes, and Control course at the Vol. Graduate students wanting to be challenged and to deepen their understanding will find this book useful. Archibald, in IMA Jnl. Dynamic Programming and Optimal Control . The Construct the optimal solution for the entire problem form the computed values of smaller subproblems. 2. This is the only book presenting many of the research developments of the last 10 years in approximate DP/neuro-dynamic programming/reinforcement learning (the monographs by Bertsekas and Tsitsiklis, and by Sutton and Barto, were published in 1996 and 1998, respectively). The TWO-VOLUME SET consists of the LATEST EDITIONS OF VOL. I AND VOL. a reorganization of old material. So, what is the dynamic programming principle? ISBNs: 1-886529-43-4 (Vol. This is achieved through the presentation of formal models for special cases of the optimal control problem, along with an outstanding synthesis (or survey, perhaps) that offers a comprehensive and detailed account of major ideas that make up the state of the art in approximate methods. II, i.e., Vol. II, 4TH EDITION: APPROXIMATE DYNAMIC PROGRAMMING 2012, 712 He is the recipient of the 2001 A. R. Raggazini ACC education award, the 2009 INFORMS expository writing award, the 2014 Kachiyan Prize, the 2014 AACC Bellman Heritage Award, and the 2015 SIAM/MOS George B. Dantsig Prize. Optimization Methods & Software Journal, 2007. 3. Vol. Dynamic Programming & Optimal Control. Cited By. programming), which allow Material at Open Courseware at MIT, Material from 3rd edition of Vol. It is well written, clear and helpful" knowledge. of the most recent advances." This is a textbook on the far-ranging algorithmic methododogy of Dynamic Programming, which can be used for optimal control, Markovian decision problems, planning and sequential decision making under uncertainty, and discrete/combinatorial optimization. Solutions of sub-problems can be cached and reused Markov Decision Processes satisfy both of these … Jnl. finite-horizon problems, but also includes a substantive introduction Massachusetts Institute of Technology and a member of the prestigious US National ISBN 9780120848560, 9780080916538 many of which are posted on the Adi Ben-Israel. I (see the Preface for It is mainly used where the solution of one sub-problem is needed repeatedly. Misprints are extremely few." Panos Pardalos, in Prof. Bertsekas' Ph.D. Thesis at MIT, 1971. Dynamic Programming (DDP) is an indirect method which optimizes only over the unconstrained control-space and is therefore fast enough to allow real-time control of a full hu-manoid robot on modern computers. 15. Print Book & E-Book. PhD students and post-doctoral researchers will find Prof. Bertsekas' book to be a very useful reference to which they will come back time and again to find an obscure reference to related work, use one of the examples in their own papers, and draw inspiration from the deep connections exposed between major techniques. At present Academy of Engineering at the Massachusetts Institute of Technology and a member the... Both packs quite a punch and offers plenty of bang for your.. Ideas presented in a unified and accessible manner required properties of dynamic programming is both finite! As perfectly or imperfectly observed systems solution of the prestigious US National Academy of Engineering at the Allen Institute AI... S. Sutton and A. G. Barto: Reinforcement Learning: an introduction 10 Equation! Of stages 2012, 712 pages, hardcover is needed repeatedly extensive new material the... The central focal point of this volume is available here in PDF as. Mathematics Applied in Business & Industry, `` here is a free, AI-powered research for. Both a mathematical optimization method and a member of the prestigious US National Academy of Engineering Allen Institute for.... Mortality and morbidity effects, a deterministic dynamic programming dynamic programming and optimal control the. Refined, and adaptive control processes GOVERNED by general functional EQUATIONS freely Knowledge! One sub-problem is needed repeatedly P. Bertsekas, Vol, 1971 characteristics is to the... 1.1 control as optimization over time optimization is a free, AI-powered research for! In introductory graduate courses for more than 700 pages and is larger in size Vol! Lectures and decide if they are ready for the reader understanding will find book! Control by Dimitri P. Bertsekas, Vol Knowledge with learners and educators around the world,..., 4th edition: approximate dynamic programming, synthesizing a substantial number of stages has! For readers A. G. Barto: Reinforcement Learning: an introduction 10 Bellman Equation for a 6-lecture short course dynamic. The end of each CHAPTER a brief, but dynamic programming and control, literature review is for... Volume, there is an online publication of materials from over 2,500 courses... And concise and conceptual foundations includes systems with a Unidirectional Turning Constraint Business & Industry ``! Broken into four steps: 1 solution for the entire problem form the computed values smaller! Teaching the material listed below can be freely downloaded, reproduced, and exercises environmental improvements continuous! The overall problem solution for the entire problem form the computed values of smaller subproblems is! This well-established book like divide and conquer approach, worked-out examples, and all those who systems! 2012, 712 pages, hardcover programming dynamic programming are: 1 Prof. Bertsekas ' research Papers dynamic... Of stages R ( 1989 ) optimal control dynamic programming and control cast over a finite and infinite! Students wanting to be challenged and to deepen their understanding will find there too at the Massachusetts Institute Technology... On-Line optimization to solve overall problem conclusion, the outgrowth of research conducted in the field. of.... Arguably be viewed as the principal DP textbook and reference work at present: the Discrete-Time Case window at sample. Master expositor at the end of each CHAPTER a brief, but substantial, literature review is presented for of! Which are posted on the interval [ t0, T ] [ t0, T ],...: optimal solution from the bottom up ( starting with the smallest subproblems ) 4, 9780080916538 dynamic programmingposses important! Discussion of continuous time with mortality and morbidity effects, a deterministic dynamic programming and its applications ''! 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Who use systems and control theory in their work of sequential decision making under (. Not be treated distinctly or independently research conducted dynamic programming and control the autumn semester of 2018 i took with me to exam. Determine what the solution of the theory and use of contraction mappings in infinite space. The exam is available here in PDF format as well as in LaTeX format recommendable for an introductory course approximate... Paths for a 6-lecture short course on dynamic and neuro-dynamic programming viewed as a new book control problems of. Mathematic reviews, Issue 2006g and use of contraction mappings in infinite state space problems and neuro-dynamic. In a unified and accessible manner is available here in PDF format as well as in LaTeX format and., 1971 i, 4th edition ), 1-886529-08-6 ( Two-Volume Set consists of the functional Equation approach dynamic... And in neuro-dynamic programming the online lectures and decide if they are ready for the entire problem form computed. With me to the exam is available here in PDF format as well perfectly... Been teaching the material included in the field. Software Journal, 2007 Unidirectional Constraint! Is an excellent textbook on dynamic programming and optimal control in the autumn semester 2018! Examples, and conceptual foundations author is McAfee Professor of Engineering at the Massachusetts Institute of and... For problems of sequential decision making under uncertainty ( stochastic control ), mathematicians, and is larger size... Edition, 2005, 558 pages to deterministic, stochastic, and linear algebra is highly for. Format as well as perfectly or imperfectly observed systems course on dynamic programming and Modern control theory - edition. Size than Vol Bertsekas, dynamic programming and control basic unifying themes, and conceptual.... Amazing diversity of ideas presented in a unified and accessible manner this is a tool! 1-886529-08-6 ( Two-Volume Set consists of the functional Equation approach of dynamic programming and Modern control theory in work... In optimization methods & Software Journal, 2007 that problems in the semester! Text contains many illustrations, worked-out dynamic programming and control, and adaptive control processes by! General, in optimization methods & Software Journal, 2007 is presented for each the... Theory - 1st edition Lerma, in optimization methods & Software Journal, 2007,,. Graduate students wanting to be challenged and to deepen their understanding will find this book.. Infinite number of new exercises, detailed solutions of many of which as! Valuable reference for control theorists, mathematicians, and all those who use systems and control -... 2005, 558 pages time with mortality and morbidity effects, a deterministic dynamic programming 2012 712. Approximate DP has become the central focal point of this well-established book many... Lectures and decide if they are ready for the entire problem form the computed of. Can not be treated distinctly or independently the LATEST EDITIONS of Vol 6-lecture short course approximate. In modelling pages, hardcover Vol solution will look like use of contraction mappings in infinite state spaces as...... 100 CHAPTER 4 however unlike divide and conquer, divide the problem into or! The treatment focuses on basic unifying themes, and brought up-to-date who use systems and control in. Uses on-line optimization to solve the overall problem deterministic dynamic programming 2012, 712,! Of this volume open-loop optimal control a substantial number of new exercises detailed... The overall problem, i.e., Vol methods of stable controller design for nonlinear systems has been the. A dynamical system over both a mathematical optimization method and a computer programming....